Problem: Simplify; express your answer in exponential form. Assume $p\neq 0, q\neq 0$. $\dfrac{{(p^{-3})^{4}}}{{(p^{-3}q^{-4})^{-5}}}$
To start, try working on the numerator and the denominator independently. In the numerator, we have ${p^{-3}}$ to the exponent ${4}$ . Now ${-3 \times 4 = -12}$ , so ${(p^{-3})^{4} = p^{-12}}$ In the denominator, we can use the distributive property of exponents. ${(p^{-3}q^{-4})^{-5} = (p^{-3})^{-5}(q^{-4})^{-5}}$ Simplify using the same method from the numerator and put the entire equation together. $\dfrac{{(p^{-3})^{4}}}{{(p^{-3}q^{-4})^{-5}}} = \dfrac{{p^{-12}}}{{p^{15}q^{20}}}$ Break up the equation by variable and simplify. $\dfrac{{p^{-12}}}{{p^{15}q^{20}}} = \dfrac{{p^{-12}}}{{p^{15}}} \cdot \dfrac{{1}}{{q^{20}}} = p^{{-12} - {15}} \cdot q^{- {20}} = p^{-27}q^{-20}$.